Clusters, Noncrossing Partitions and the Coxeter Plane Nathan Reading

The present research is a contribution to W -Catalan combinatorics, which concerns generalizations of objects counted by the Catalan number to an arbitrary finite Coxeter group W. Triangulations of a polygon are a special case of the combinatorial clusters appearing in the theory of cluster algebras of finite type. Planar diagrams analogous to triangulations encode the combinatorics of clusters in the classical types, but no planar models (or other combinatorial models) are known for clusters associated to exceptional finite Coxeter groups. The usual noncrossing partitions can be seen as encoding the combinatorics of a certain nonstandard presentation of the symmetric group, and noncrossing partitions for any finite Coxeter group can be defined by an analogous construction. However, planar diagrams for these generalized noncrossing partitions are unknown outside of the classical types. In this extended abstract, we report on work in progress to construct planar diagrams for clusters and noncrossing partitions of arbitrary finite type.